One of the principal distinctions between ordinary light and laser light is related to beam propagation. Ordinary light may be thought of as the aggregate sum of a large number of individual spherical electromagnetic waves each radiating from its own separate point source (called Huygen's wavelets in optics textbooks). The individual sources making up the emitter act independently without correlation in space, time or frequency between the individual waves. A familiar consequence, taught in traditional optics texts, is that if one uses a lens to collect such light and concentrate it as tightly as possible in the lens focal plane, the best result is an illuminated region that is the composite image of all the individual incoherent sources with a surface brightness no greater than the surface brightness of the source. The size of this illuminated area will grow in proportion to the size of the emitter and any attempt to create higher intensity in the focal spot by increasing the size (and total power radiated) of the emitter will prove unsuccessful. In contrast, laser light can be characterized as a single wavetrain (or a small number of wavetrains) having uniform frequency and phase over a broad space transverse to the propagation direction. Such coherent light can be concentrated by a lens to a diameter that can be as small as the Heisenberg uncertainty principle allows (i.e., focused to the "diffraction limit"), regardless of the spatial extent of the source.
The way in which intensity is distributed across the laser beam has important consequences in the degree to which a small focal-spot diameter can be achieved. A beam comprising a single wavetrain with a Gaussian intensity distribution (shown as the TEM.sub.00 mode profile in FIG. 1) will focus more tightly than any other distribution. Such Gaussian beams can be produced by lasers, but typically at a significant sacrifice in output power. Lasers employ resonant cavities to store optical energy, and this energy is stored in various "modes", each with its own frequency and distribution of intensity. The mode of lowest order is the one that has the Gaussian intensity distribution just described, and is for that same reason, the mode that occupies the smallest diameter inside the laser resonator. Higher order modes have more complex intensity distributions and occupy larger cross-sectional areas.
The mode profiles of FIG. 1 are used as examples throughout and so are here briefly explained. For each mode, the intensity in the beam (vertical axis) is plotted as a function of the radial distance outward (horizontal axis) transverse to the beam propagation axis. This intensity could be measured, for example, by detecting the power transmitted through a small pinhole aperture (one with an opening small compared to the beam width) as the pinhole was moved across the beam perpendicular to the propagation axis. The characteristic radius of the TEM.sub.00 mode is used for the units for the horizontal axis, where the definition of this characteristic radius is the conventional "1/e.sup.2 radius" as explained later. The mode designations TEM.sub.pl are given in cylindrical coordinates, for radially symmetric intensity profiles (there are similar-looking, but different designations for rectangular coordinates, so the coordinate system must be specified). The units of the intensity axis are relative but normalized so that the same power is contained in each of the modes. The mode order--essentially the number of nodes in the profile, which determines the rate at which the beam spreads--increases by one with each plot, stepping down in FIG. 1, the TEM.sub.00 mode being the lowest order mode, TEM.sub.01 * being the next highest, etc.; these are the six cylindrical modes of lowest order. The modes are shown with the correct relative scales to be a set of higher order modes generated by one laser, in one resonator. A mixed mode profile is made up by summing these profiles (or even higher order profiles) with weight factors proportional to the relative power of that mode in the mixture. For example, adding the first three profiles together would make a mode profile that peaked in the center and was without radial nodes; this would resemble the TEM.sub.00 mode but would always propagate with a larger diameter. Modes are useful concepts not only because these intensity profiles are generated by real lasers but because these profiles (including mixed mode profiles) remain unchanged as the laser beam propagates or is focused. Since the same profiles apply at the focus of a lens, the figure indicates the relative scale of how tightly each mode from a given laser can be focused.
It is generally possible to constrain any laser to operate only in the fundamental (TEM.sub.00) mode by reducing the diameter of the limiting aperture in the laser cavity, thereby increasing the optical loss such that no mode except the TEM.sub.00 mode experiences sufficient net gain to oscillate. The price for this is lower power in the output beam than if several modes were oscillating. Laser manufacturers usually choose to compromise between the two competing goals, high power and TEM.sub.00 mode, by using designs that allow some power in the higher order modes. The intensity distributions for these mixtures of modes can appear to be Gaussian, but they do not propagate or focus with as small a transverse extent as the pure TEM.sub.00 mode would.
The fact that a laser beam can have a Gaussian-like intensity distribution, but yet have the higher power possible only with a mixed-modes, has led some laser manufacturers to improperly claim that their products generate pure TEM.sub.00 beams. In fact their products produce beams that are a mixture of higher order modes, with an intensity profile that only superficially resembles a fundamental mode profile. This has lead to unrealized expectations, often with great economic consequences, for unwary designers of laser systems and naive laser users. An important factor which has allowed these problems to continue has been the absence of a simple, easy-to-use, low-cost instrument for quantifying beam quality in a meaningful way. An even more fundamental problem has been the lack of a theoretical basis for a meaningful way to quantify beam quality. A related issue is that while there exists much information about how to compute the propagation of TEM.sub.00 beams, there is no practical set of analytical "tools" with which to predict the attributes of a mixed-mode beam as it propagates through an optical system. It is these analytical tools and the resulting instrumentation for quantifying beam quality which are the subjects of this invention.
The absence of an analytical description for the propagation of the mixtures of higher order modes that come out of real lasers makes it very difficult to design optical beam delivery systems which must channel these multimode beams through a plurality of optical elements to a workpiece. In an industrial laser beam delivery system, selecting the proper elements is extremely important. For example, the final delivery lens is typically very close to the workpiece, and often becomes contaminated from material removed from the workpiece. Due to this contamination, the final delivery lens must be replaced very frequently, sometimes many times a day. These lenses can be quite expensive. Moreover, the cost of these lenses goes up significantly with their size. Accordingly it is desirable to design a beam delivery system where the optical elements are as small as possible while still being capable of passing the beam. This requires knowledge of the propagation characteristics of the higher order mode beam. It would be best to have an analytical model that would show how to optimally prepare the beam out of the laser for launch into the delivery system so as to minimize the cost of the whole system.
At present, there are only very crude methods available for designing optical delivery systems for industrial lasers. In one approach, the laser beam is directed across a room, over a distance equivalent to that which will be used in the commercial application. This distance can be many meters. Technicians will then insert pieces of plastic at various locations in the beam path. The burn hole created in each piece of plastic can be used to give information about the diameter of the beam and the intensity distribution pattern at that location. In addition to being crude, this latter approach is dangerous because the technician and other employees must move around a room though which a mulitkilowatt invisible beam is being transmitted. It would clearly be desirable to have an improved way of determining the mode quality or mixture of modes in a laser beam.
As noted above, most laser beams, and particularly those generated by high power commercial lasers, are comprised of a mixture of higher order modes. For the purposes of designing optical elements, it has been found that a detailed knowledge of all the underlying modes is unnecessary. Rather, as will be shown, the propagation of a multimode laser beam through an optical system can be predicted if one can characterize the beam by a numerical "quality". This quality number will turn out to be the same figure as "times-diffraction-limit" figure known in optics literature and it is the factor by which the focus-spot diameter for a high-order beam is larger than that for a TEM.sub.00 beam having the same diameter at the focusing optic. Equivalently, it is the factor by which the angle of spreading, or divergence angle, for a high order mode is increased over a diffraction-limited, TEM.sub.00 mode beam of the same waist diameter.
FIG. 2 illustrates a high order mode or multimode laser beam 10 propagating along an axis 12. The beam converges to a smallest diameter 14 (perpendicular to the axis) called the waist of value 2W.sub.o, and then diverges propagating away from the waist location. In a distance of propagation Z.sub.R on either side of the waist called the Rayleigh range 16, the beam diameter is larger by a factor of .sqroot.2 than the waist diameter. This means the beam cross-sectional area has doubled.
Within this beam is drawn a representation of the beam propagation of the fundamental mode 18 of Gaussian intensity profile which has the same waist location as the multimode beam 10, and a Rayleigh range 20 (propagation distance for the area of this beam to double) given the lower case symbol z.sub.R of the same value as beam 10, or z.sub.R =Z.sub.R. The beam 18 with these properties will be termed the "associated fundamental mode" for the multimode beam 10. The laws of diffraction dictate that a beam of waist diameter A will spread at large distances from the waist with a divergence angle inversely proportional to the waist diameter and proportional to the wavelength .lambda. of the light, or EQU (divergence)=(constant)(.lambda./A) (1)
The proportionality constant depends on the intensity distribution across the beam and the way the beam diameter is defined. The smallest possible constant occurs for a fundamental mode with a Gaussian intensity profile I(r,z) given by EQU I(r,z)=I.sub.o exp[-2r2/(w(z))].sup.2 ( 2)
where I.sub.o =2P/xw2 and is the intensity at the center of the beam, P is the total power in the beam, r is the radial distance from the center and w(z) is a radial scale parameter which increases with distance z away from the waist.
If the beam diameter is taken as twice the radius for which the intensity has dropped to 1/e.sup.2 =13.5% of the central intensity, then 2w(z) is the beam diameter at distance z, and the constant in the divergence expression for this fundamental mode is 4/.pi.. The laws of diffraction relate the Rayleigh range for a fundamental mode to the waist radius by the expression: EQU z.sub.R =.pi.w.sub.o.sup.2 /.lambda. (3)
so that for the beam 18 the waist diameter 22 is equal to: EQU 2w.sub.o =2(.lambda.z.sub.R /.pi.).sup.1/2 ( 4)
and the far field divergence angle 24 is EQU .theta..sub.F =(4/.lambda.)(.pi./2w.sub.o) or 4.lambda./(.lambda.2w.sub.o). (5)
Equation (5) is equivalent to equation (1) with the "constant" expressed as (4/.pi.).
The divergence angle .THETA..sub.F (shown at 26 in FIG. 2) for the multimode beam 10 is greater than for the associated fundamental mode beam 18 by a constant factor M equal to: EQU .THETA..sub.F =M.theta..sub.F ( 6)
where the value of M depends on the definition of beam diameter used for the multimode beam. Given these definitions, the laws of light propagation give that the beam diameter 2W(z) for beam 10 is everywhere just M times larger than that for the associated fundamental mode, or EQU W(z)=Mw(z) (7)
and in particular the waists of the two beams are related by EQU 2W.sub.o =2Mw.sub.o. (8)
To characterize the propagation of the multimode beam, the factor by which its divergence angle, .THETA..sub.F, exceeds the divergence angle of a TEM.sub.00 beam having the same waist diameter must be determined. A TEM.sub.00 mode having a waist diameter of 2W.sub.o will diverge more slowly than the smaller associated fundamental mode according to: EQU 4.lambda./.pi.(2W.sub.o)=4.lambda./.pi.M(2w.sub.o)=.theta..sub.F /M (9)
The "quality" of the multimode beam is just the ratio of its divergence, M.theta..sub.F to that for a TEM.sub.00 of the same waist diameter expressed in (9). This ratio is seen to be M.THETA..sub.F /(.theta..sub.F /M) or M.sup.2. Thus, M.sup.2 is the quantity which characterizes the quality of beam 10 wherein smaller values represent a higher beam quality.
This definition leads to a number of observations. First, a beam that is a pure fundamental mode has an M.sup.2 value of 1. Second, as the value of M.sup.2 increases, the divergence of the beam increases and quality decreases. Most importantly, analytical tools can be developed for predicting the propagation of a multimode laser beam if the value of M.sup.2, 2W.sub.o and the waist location are known. (The equation for beam propagation of a higher order mode beam is shown below at (11).)
All modes generated in a given laser resonator, have the same radii of curvature and propagate with the same characteristic distance (Rayleigh range) for the beam area to double, regardless of the mode order, as is well known in the literature (see e.g., H. Koglenik and T. Li, Applied Optics, Vol. 5, October 1966, pages 1550-1567). The associated fundamental mode was defined by matching its Rayleigh range to that of the multimode beam. This means the associated fundamental mode found above is the same TEM.sub.00 mode that would be generated and appear in the ensemble of modes comprising a multimode beam for that resonator. Each pure higher-order mode making up the multimode beam is everywhere along the z-axis, larger than this underlying TEM.sub.00 mode by a constant factor; and so the diameter of the entire sum-of-modes is everywhere larger by the constant factor M. This makes Equation (7) true for multimode laser beams, and by so identifying the associated fundamental mode, a propagation law (equation 11) for the multimode beam is obtained by appropriate insertion of the M-factor in the well-known propagation law for the fundamental mode.
To identify the associated fundamental mode, the Rayleigh range of the multimode beam must be measured, requiring measurement of the location of the beam waist, and a minimum of two beam diameters at known distances from the waist. When this data is fitted to the modified propagation law to give the values of 2Wo, M.sup.2, and the location Z.sub.o of the beam waist, the desired analytical model of multimode beam propagation is achieved. Equation 11 becomes the mathematical tool that provides the ability to model and predict beam diameter and wavefront curvature at any location in an optical system for multimode beams. Ray matrix methods, for example, become as useful for high mode-order beams as Koglenik and Li showed them to be in the reference cited above for fundamental mode beams.
While it is possible to predict the propagation of a laser beam if the value of M.sup.2, waist diameter and location can be determined, there is no device presently existing which can adequately measure these parameters. At best, researchers have been limited to measuring the profiles of beams at a plurality of locations to obtain an empirical feel for mode quality. Information about beam profiles can be obtained by burning plastic blocks as discussed above. Much more sophisticated beam profilers are available which can give information about the energy distribution of a beam at a location in space. However, information about beam diameter or energy distribution at a one location does not provide enough data to derive a value for M.sup.2 or the other parameters. Therefore it would be desirable to have an apparatus which can directly measure the model parameters for a multimode laser beam.
An apparatus for measuring the mode quality of a beam will have many uses beyond designing beam delivery systems. For example, many scientific experiments require that the beam quality be at or near the unity value of the fundamental mode. Such an apparatus can be used to measure beam quality as the laser resonator mirrors are peaked in angular alignment so that the TEM.sub.00 mode output can be maximized.
Other uses of the subject invention will include the pinpointing of misadjusted or imperfect elements in an optical train. More specifically, since the quality of a beam will be degraded when passed through an imperfect optical element, by comparing the beam quality both before and after passing through a particular optic, information can be derived about the optic itself. The subject apparatus can also be helpful in reproducing experimental results that depend on the mode mixture of the input multimode beam.
Accordingly, it is an object of the subject invention to provide a method and apparatus for determining the quality of a laser beam.
It is another object of the subject invention to provide an apparatus which can determine beam quality at a single location near the output of the laser.
It is still another object of the subject invention to provide a method for optimum design of laser beam delivery systems based on knowledge of the mode quality of a laser beam.
It is still a further object of the subject invention to provide an apparatus which can generate information about the alignment of a laser beam.
It is still another object of the subject invention to provide an apparatus which generates information about the pointing stability of a laser.
It is still a further object of the subject invention to provide an apparatus for generating improved beam profiles.
It is still another object of the subject invention to provide an apparatus which can generate information about the underlying modes forming a multimode laser beam.